In this post, I introduce the concept of optional and predictable projections of jointly measurable processes. Optional projections of right-continuous processes and predictable projections of left-continuous processes were constructed in earlier posts, with the respective continuity conditions used to define the projection. These are, however, just special cases of the general theory. For arbitrary measurable processes, the projections cannot be expected to satisfy any such pathwise regularity conditions. Instead, we use the measurability criteria that the projections should be, respectively, optional and predictable.

The projection theorems are a relatively straightforward consequence of optional and predictable section. However, due to the difficulty of proving the section theorems, optional and predictable projection is generally considered to be an advanced or hard part of stochastic calculus. Here, I will make use of the section theorems as stated in an earlier post, but leave the proof of those until after developing the theory of projection.

As usual, we work with respect to a complete filtered probability space, and only consider real-valued processes. Any two processes are considered to be the same if they are equal up to evanescence. The optional projection is then defined (up to evanescence) by the following.

Theorem 1 (Optional Projection) Let X be a measurable process such that is almost surely finite for each stopping time. Then, there exists a unique optional process , referred to as the optional projection of X, satisfying

(1)

almost surely, for each stopping time .

Predictable projection is defined similarly.

Theorem 2 (Predictable Projection) Let X be a measurable process such that is almost surely finite for each predictable stopping time. Then, there exists a unique predictable process , referred to as the predictable projection of X, satisfying

As I have mentioned before in these notes, when working with processes in continuous time, it is important to select a good modification. Typically, this means that we work with processes which are left or right continuous. However, in general, it can be difficult to show that the paths of a process satisfy such pathwise regularity. In this post I show that for optional and predictable processes, the section theorems introduced in the previous post can be used to considerably simplify the situation. Although they are interesting results in their own right, the main application in these notes will be to optional and predictable projection. Once the projections are defined, the results from this post will imply that they preserve certain continuity properties of the process paths.

Suppose, for example, that we have a continuous-time process X which we want to show to be right-continuous. It is certainly necessary that, for any sequence of times decreasing to a limit , almost-surely tends to . However, even if we can prove this for every possible decreasing sequence , it does not follow that X is right-continuous. As a counterexample, if is any continuously distributed random time, then the process is not right-continuous. However, so long as the distribution of has no atoms, X is almost-surely continuous at each fixed time t. It is remarkable, then, that if we generalise to look at sequences of stopping times, then convergence in probability along decreasing sequences of stopping times is enough to guarantee everywhere right-continuity of the process. At least, it is enough so long as we restrict consideration to optional processes.

As usual, we work with respect to a complete filtered probability space. Two processes are considered to be the same if they are equal up to evanescence, and any pathwise property is said to hold if it holds up to evanescence. That is, a process is right-continuous if and only is it is everywhere right-continuous on a set of probability 1. All processes will be taken to be real-valued, and a process is said to have left (or right) limits if its left (or right) limits exist everywhere, up to evanescence, and are finite.

X is right-continuous if and only if in probability, for each uniformly bounded sequence of stopping times decreasing to a limit .

X has right limits if and only if converges in probability, for each uniformly bounded decreasing sequence of stopping times.

X has left limits if and only if converges in probability, for each uniformly bounded increasing sequence of stopping times.

The `only if’ parts of these statements is immediate, since convergence everywhere trivially implies convergence in probability. The importance of this theorem is in the `if’ directions. That is, it gives sufficient conditions to guarantee that the sample paths satisfy the respective regularity properties.

Note that conditions for left-continuity are absent from the statements of Theorem 1. In fact, left-continuity does not follow from the corresponding property along sequences of stopping times. Consider, for example, a Poisson process, X. This is right-continuous but not left-continuous. However, its jumps occur at totally inaccessible times. This implies that, for any sequence of stopping times increasing to a finite limit , it is true that converges almost surely to . In light of such examples, it is even more remarkable that right-continuity and the existence of left and right limits can be determined by just looking at convergence in probability along monotonic sequences of stopping times. Theorem 1 will be proven below, using the optional section theorem.

Consider a probability space and a subset S of . The projection is the set of such that there exists a with . We can ask whether there exists a map

such that . From the definition of the projection, values of satisfying this exist for each individual . By invoking the axiom of choice, then, we see that functions with the required property do exist. However, to be of use for probability theory, it is important that should be measurable. Whether or not there are measurable functions with the required properties is a much more difficult problem, and is answered affirmatively by the measurable selection theorem. For the question to have any hope of having a positive answer, we require S to be measurable, so that it lies in the product sigma-algebra , with denoting the Borel sigma-algebra on . Also, less obviously, the underlying probability space should be complete. Throughout this post, will be assumed to be a complete probability space.

It is convenient to extend to the whole of by setting for outside of . Then, is a map to the extended nonnegative reals for which precisely when is in . Next, the graph of , denoted by , is defined to be the set of with . The property that whenever is expressed succinctly by the inclusion . With this notation, the measurable selection theorem is as follows.

Theorem 1 (Measurable Selection) For any , there exists a measurable such that and

(1)

As noted above, if it wasn’t for the measurability requirement then this theorem would just be a simple application of the axiom of choice. Requiring to be measurable, on the other hand, makes the theorem much more difficult to prove. For instance, it would not hold if the underlying probability space was not required to be complete. Note also that, stated as above, measurable selection implies that the projection of S is equal to a measurable set , so the measurable projection theorem is an immediate corollary. I will leave the proof of Theorem 1 for a later post, together with the proofs of the section theorems stated below.

A closely related problem is the following. Given a measurable space and a measurable function, , does there exist a measurable right-inverse on the image of ? This is asking for a measurable function, , from to such that . In the case where is the Borel space , Theorem 1 says that it does exist. If S is the graph then will be the required right-inverse. In fact, as all uncountable Polish spaces are Borel-isomorphic to each other and, hence, to , this result applies whenever is a Polish space together with its Borel sigma-algebra. (more…)

In contrast to optional processes, the class of predictable processes was used extensively in the development of stochastic integration in these notes. They appeared as integrands in stochastic integrals then, later on, as compensators and in the Doob-Meyer decomposition. Since they are also central to the theory of predictable section and projection, I will revisit the basic properties of predictable processes now. In particular, any of the collections of sets and processes in the following theorem can equivalently be used to define the predictable sigma-algebra. As usual, we work with respect to a complete filtered probability space. However, completeness is not actually required for the following result. All processes are assumed to be real valued, or take values in the extended reals .

Theorem 1 The following collections of sets and processes each generate the same sigma-algebra on .

Compare this with the analogous result for sets/processes generating the optional sigma-algebra given in the previous post. The proof of Theorem 1 is given further below. First, recall that the predictable sigma-algebra was previously defined to be generated by the left-continuous adapted processes. However, it can equivalently be defined by any of the collections stated in Theorem 1. To make this clear, I now restate the definition making use if this equivalence.

Definition 2 The predictable sigma-algebra, , is the sigma-algebra on generated by any of the collections of sets/processes in Theorem 1.

15 November 16

The optional sigma-algebra, , was defined earlier in these notes as the sigma-algebra generated by the adapted and right-continuous processes. Then, a stochastic process is optional if it is -measurable. However, beyond the definition, very little use was made of this concept. While right-continuous adapted processes are optional by construction, and were used throughout the development of stochastic calculus, there was no need to make use of the general definition. On the other hand, optional processes are central to the theory of optional section and projection. So, I will now look at such processes in more detail, starting with the following alternative, but equivalent, ways of defining the optional sigma-algebra. Throughout this post we work with respect to a complete filtered probability space, and all stochastic processes will be assumed to be either real-valued or to take values in the extended reals .

Theorem 1 The following collections of sets and processes each generate the same sigma-algebra on .

The optional-sigma algebra was previously defined to be generated by the right-continuous adapted processes. However, any of the four collections of sets and processes stated in Theorem 1 can equivalently be used, and the definitions given in the literature do vary. So, I will restate the definition making use of this equivalence.

Definition 2 The optional sigma-algebra, , is the sigma-algebra on generated by any of the collections of sets/processes in Theorem 1.

Theorem 1 seems almost obvious. Projection is a very simple map and we may well expect the projection of, say, a Borel subset of onto to be Borel. In order to formalise this, we could start by noting that sets of the form for Borel A and B have an easily described, and measurable, projection, and the Borel sigma-algebra is the closure of the collection such sets under countable unions and under intersections of decreasing sequences of sets. Furthermore, the projection operator commutes with taking the union of sequences of sets. Unfortunately, this method of proof falls down when looking at the limit of decreasing sequences of sets, which does not commute with projection. For example, the decreasing sequence of sets all project onto the whole of , but their limit is empty and has empty projection.

There is an interesting history behind Theorem 1, as mentioned by Gerald Edgar on MathOverflow (1) in answer to The most interesting mathematics mistake? In a 1905 paper, Henri Lebesgue asserted that the projection of a Borel subset of the plane onto the line is again a Borel set (Lebesgue, (3), pp 191–192). This was based on the erroneous assumption that projection commutes with the limit of a decreasing sequence of sets. The mistake was spotted, in 1916, by Mikhail Suslin, and led to his investigation of analytic sets and to begin the study of what is now known as descriptive set theory. See Kanamori, (2), for more details. In fact, as was shown by Suslin, projections of Borel sets need not be Borel. So, by considering the case where and , Theorem 1 is false if the completeness assumption is dropped. I will give a proof of Theorem 1 but, as it is a bit involved, this is left for a later post.

For now, I will state some consequences of the measurable projection theorem which are important to the theory of continuous-time stochastic processes, starting with the following. Throughout this post, the underlying probability space is assumed to be complete, and stochastic processes are taken to be real-valued, or take values in the extended reals , with time index ranging over . For a first application of measurable projection, it allows us to show that the supremum of a jointly measurable processes is measurable.

Lemma 2 If X is a jointly measurable process and then is measurable.

Proof: Setting then, for each real K, if and only if for some . Hence,

By the measurable projection theorem, this is in and, as sets of the form generate the Borel sigma-algebra on , U is -measurable. ⬜

Next, the running maximum of a jointly measurable process is again jointly measurable.

Lemma 3 If X is a jointly measurable process then is also jointly measurable.

In the previous post, I looked at optional projection. Given a non-adapted process X we construct a new, adapted, process Y by taking the expected value of conditional on the information available up until time t. I will now concentrate on predictable projection. This is a very similar concept, except that we now condition on the information available strictly before time t.

It will be assumed, throughout this post, that the underlying filtered probability space satisfies the usual conditions, meaning that it is complete and right-continuous. This is just for convenience, as most of the results stated here extend easily to non-right-continuous filtrations. The sigma-algebra

represents the collection of events which are observable before time t and, by convention, we take . Then, the conditional expectation of X is written as,

(1)

By definition, Y is adapted. However, at each time, (1) only defines Y up to a zero probability set. It does not determine the paths of Y, which requires specifying its values simultaneously at the uncountable set of times in . So, (1) does not tell us the distribution of Y at random times, and it is necessary to specify an appropriate version for Y. Predictable projection gives a uniquely defined modification satisfying (1). The full theory of predictable projection for jointly measurable processes requires the predictable section theorem. However, as I demonstrate here, in the case where X is left-continuous, predictable projection can be done by more elementary methods. The statements and most of the proofs in this post will follow very closely those given previously for optional projection. The main difference is that left and right limits are exchanged, predictable stopping times are used in place of general stopping times, and the sigma algebra is used in place of .

Stochastic processes will be defined up to evanescence, so two processes are considered to be the same if they are equal up to evanescence. In order to apply (1), some integrability requirements need to imposed. I will use local integrability. Recall that, in these notes, a process X is locally integrable if there exists a sequence of stopping times increasing to infinity and such that

(2)

is integrable. This is a strong enough condition for the conditional expectation (1) to exist, not just at each fixed time, but also whenever t is a stopping time. The main result of this post can now be stated.

As it is left-continuous, the fact that Y is specified, almost surely, at any time t by (1) means that it is uniquely determined up to evanescence. The main content of Theorem 1 is the existence of Y, and the proof of this is left until later in this post.

The process defined by Theorem 1 is called the predictable projection of X, and is denoted by . So, is the unique left-continuous process satisfying

(3)

for all times t. In practice, X will usually not just be left-continuous, but will also have right limits everywhere. That is, it is caglad (“continu à gauche, limites à droite”).

The simplest non-trivial example of predictable projection is where is constant in t and equal to an integrable random variable U. Then, is the left-limits of the cadlag martingale , so is easily seen to be a caglad process. (more…)

In filtering theory, we have a filtered probability space and a signal process. The sigma-algebra represents the collection of events which are observable up to and including time t. The process X is not assumed to be adapted, so need not be directly observable. For example, we may only be able to measure an observation process, which incorporates some noise , and generates the filtration , so is adapted. The problem, then, is to compute an estimate for based on the observable data at time t. Looking at the expected value of X conditional on the observable data, we obtain the following estimate for X at each time ,

(1)

The process Y is adapted. However, as (1) only defines Y up to a zero probability set, it does not give us the paths of Y, which requires specifying its values simultaneously at the uncountable set of times in . Consequently, (1) does not tell us the distribution of Y at random times. So, it is necessary to specify a good version for Y.

Optional projection gives a uniquely defined process which satisfies (1), not just at every time t in , but also at all stopping times. The full theory of optional projection for jointly measurable processes requires the optional section theorem. As I will demonstrate, in the case where X is right-continuous, optional projection can be done by more elementary methods.

Throughout this post, it will be assumed that the underlying filtered probability space satisfies the usual conditions, meaning that it is complete and right-continuous, . Stochastic processes are considered to be defined up to evanescence. That is, two processes are considered to be the same if they are equal up to evanescence. In order to apply (1), some integrability requirements need to imposed on X. Often, to avoid such issues, optional projection is defined for uniformly bounded processes. For a bit more generality, I will relax this requirement a bit and use prelocal integrability. Recall that, in these notes, a process X is prelocally integrable if there exists a sequence of stopping times increasing to infinity and such that

(2)

is integrable. This is a strong enough condition for the conditional expectation (1) to exist, not just at each fixed time, but also whenever t is a stopping time. The main result of this post can now be stated.

Back when I first started this series of posts on stochastic calculus, the aim was to write up the notes which I began writing while learning the subject myself. The idea behind these notes was to give a more intuitive and natural, yet fully rigorous, approach to stochastic integration and semimartingales than the traditional method. The stochastic integral and related concepts were developed without requiring advanced results such as optional and predictable projection or the Doob-Meyer decomposition which are often used in traditional approaches. Then, the more advanced theory of semimartingales was developed after stochastic integration had already been established. This now complete! The list of subjects from my original post have now all been posted. Of course, there are still many important areas of stochastic calculus which are not adequately covered in these notes, such as local times, stochastic differential equations, excursion theory, etc. I will now focus on the projection theorems and related results. Although these are not required for the development of the stochastic integral and the theory of semimartingales, as demonstrated by these notes, they are still very important and powerful results invaluable to much of the more advanced theory of continuous-time stochastic processes. Optional and predictable projection are often regarded as quite advanced topics beyond the scope of many textbooks on stochastic calculus. This is because they require some descriptive set theory and, in particular, some understanding of analytic sets. The level of knowledge required for applications to stochastic calculus is not too great though, and I aim to give complete proofs of the projection theorems in these notes. However, the proofs of these theorems do require ideas which are not particularly intuitive from the viewpoint of stochastic calculus, and hence the desire to avoid them in the initial development of the stochastic integral. The theory of semimartingales and stochastic integration will not used at all in the series of posts on the projection theorems, and all that will be required from these stochastic calculus notes are the initial posts on filtrations and processes. I will also mention quasimartingales, although only the definition and very basic properties will be required.

The subjects related to the projection theorems which I will cover are,

The Debut Theorem. I have already covered the debut theorem for right-continuous processes. This is a special case of the more general result which applies to arbitrary progressively measurable processes.

Optional and Predictable Projection. This forms the core of these sequence of posts, and follows in a straightforward way from the section theorems. As the section theorems are required to prove them, the projection theorems are also regarded as an advanced topic. However, for right-continuous and left-continuous processes it is possible to construct respectively the optional and predictable projections in a more elementary and natural way, without involving the section theorems.

Dual Optional and Predictable Projection. The dual projections are, as the name suggests, dual to the optional and predictable projections mentioned above. These apply to increasing integrable processes or, more generally, to processes with integrable variation. For a process X, the dual projections can be thought of as the optional and predictable projections applied to the differential .

The Doléans Measure. The Doléans measure can be defined for class (D)submartingales and, applied to the square of a martingale, can be used to construct the stochastic integral for square integrable martingales. Although this does not involve the projection theorems, the Doléans measure in conjunction with dual predictable projection gives a slick proof of the Doob-Meyer decomposition. The Doléans measure also exists for quasimartingales and, similarly, the Doob-Meyer decomposition can be extended to such processes.